asymmetric treatment effects

[Xinya Hao]

Hi everyone!

Have a good day~

If I believe that the impact of the rise and fall of X on Y is asymmetric, how should I design my empirical study?

Here is my story. (It's a bit long.) Thank you for your attention and any possible comments and suggestions!

I am attempting to analyze whether changes in tax rates (X) determined by the city governments affect firm performance (Y).
My theoretical analysis suggests that an increase in X would decrease Y, whereas a decrease in X would not necessarily lead to an increase in Y.
So, as the title of the question, the impact of the rise and fall of X is asymmetric.

In the past few years, there has been a lot of discussion about heterogeneous treatment effects, especially the treatment effects of different timings.
However, there doesn't seem to be much progress in thinking about asymmetric treatment effects.

I have collected the following data:
- Tax rates at the city level (panel data for hundreds of cities) (and other city level characteristics)
- Firm-level data (panel data)

Considering several factors, some of which are unobservable, which may simultaneously influence X (in terms of magnitude and timing of tax rate changes) and Y, I am using the Difference-in-Differences (DID) method to alleviate endogeneity concerns. My designed model is as follows:

1. Effect of tax rate increase:
Sample: Firms in cities where tax rates have remained unchanged and firms in cities that experienced one/multiple increase(s) in tax rates. Firms in cities where tax rates have decreased once or multiple times are excluded.
(1) Y_it = a₁ + b₁*X_it + firm FE + year FE + error
The standard errors are clustered at the city level.

2. Effect of tax rate decrease:
Sample: Similar to the previous one, but excludes firms in cities where tax rates have increased once or multiple times.
(2) Y_it = a₂ + b₂*X_it + firm FE + year FE + error

I hope that models (1) and (2) make sense.

However, I am considering other possible approaches because:
- Models (1) and (2) use different samples, although they are based on the same control group sample. Could this be a problem?
- A significant amount of data is not utilized since some cities have experienced both increases and decreases (in different years. The tax rate only changes once in a given year.) in X.
- After estimating models (1) and (2), I may need to further test b1 + b2 = 0.

3. Simultaneously estimate the impact of tax rate increase and decrease on Y.
Define:
xup = x_it – x_it-1 if x_it > x_it-1, xup = 0 otherwise;
xdown = |x_it – x_it-1| if x_it < x_it-1, xdown = 0 otherwise;

Can I estimate the model:
(3) Y_it = X_it-1 + b₁*xup + b₂*xdown + e [ considering that X_it = X_it-1 + (X_it - X_it-1) ]
to achieve my goal? Is model (3) impossible? If it is possible, under what conditions?

Any ideas, discussions, suggestions, and criticisms are welcome.

Kind regards,
Hall

[Clyde Schechter]

I would definitely go with model 3. I'm puzzled that you ask whether it is impossible. Why do you think it might be impossible? I can't think of any conditions under which it would be impossible.

The approach based on 1) and 2) looks seriously problematic to me. Cities that only increase taxes, or only decrease taxes during your observation period, and never have them oscillate, will be a biased sample. It's also not clear to me what the control group in your analysis would be. Perhaps in 1) it would be cities that had no tax change at all, or perhaps it would cities that had either no tax change or a decrease? Analogously in 2. Either way, the control groups in the two analyses would overlap, which makes it difficult (actually, as far as I know, impossible) to then do a valid comparison of coefficients between the models.

Finally, I can't help remarking the the potential for endogeneity in this process boggles my mind, and I am not persuaded that just doing a DID analysis will be sufficient to identify a causal effect here. But I'm no economist, so I'll leave that to your judgment.

[Jared Greathouse]

Perhaps a synthetic instrument is in order (not something you can do in Stata unfortunately, you'd need to code it).

[Xinya Hao]

I would definitely go with model 3. I'm puzzled that you ask whether it is impossible. Why do you think it might be impossible? I can't think of any conditions under which it would be impossible.

The approach based on 1) and 2) looks seriously problematic to me. Cities that only increase taxes, or only decrease taxes during your observation period, and never have them oscillate, will be a biased sample. It's also not clear to me what the control group in your analysis would be. Perhaps in 1) it would be cities that had no tax change at all, or perhaps it would cities that had either no tax change or a decrease? Analogously in 2. Either way, the control groups in the two analyses would overlap, which makes it difficult (actually, as far as I know, impossible) to then do a valid comparison of coefficients between the models.

Finally, I can't help remarking the the potential for endogeneity in this process boggles my mind, and I am not persuaded that just doing a DID analysis will be sufficient to identify a causal effect here. But I'm no economist, so I'll leave that to your judgment.

Thanks for your advice.

In model 1), the sample includes the firms in cities that only experience tax increases and no tax change at all (control group).
In model 2), the sample includes the firms in cities that only experience tax decreases and no tax change at all (control group).

Do you suggest that there are some prior factors that have led the city government to make the choice of continuously increasing or reducing tax rates? Consequently, if we claim that the effects of the tax increase and decrease estimated by models (1) and (2) are asymmetric, it is problematic as they only reflect the differences between the cities before the tax rate changes. In other words, I cannot rule out the possibility that the impact of tax rate changes is indeed symmetrical, but only the impact of tax rates is heterogeneous in different cities.

But will this problem remain in the model (3)?

BTW, I thought using an overlapped control group makes the comparison coefficients possible. Why this is impossible?

[Clyde Schechter]

But will this problem remain in the model (3)?

Yes. Moreover, it will remain with any analysis of this data. To overcome this you need a design that also incorporates information about these preceding factors, either directly or indirectly.

BTW, I thought using an overlapped control group makes the comparison coefficients possible. Why this is impossible?

I misspoke. I was thinking of a specific analysis that would fail in the presence of overlapping controls, but there are other approaches possible.

[Jeff Wooldridge]

One reason I’d use the third option is that using two samples means you’re effectively changing the firm fixed effect for those firms appearing as the controls. They seems a bit odd to me. As long as you’re assuming the tax variable is strictly exogenous with respect to the shocks, choosing the data on the basis of the tax variable doesn’t cause bias. But I still think interacting the change with dummy variables indicating a negative or positive change is the best bet.

[Xinya Hao]

Thank you for your responses. I have been pondering this question for a few days.

I will take a look at the synthetic instrument. It is something totally new for me.

At the very least, I now realize that it is okay to split the variation of a variable into two components for analysis.
However, I am currently concerned that the changes in tax rates may not be entirely exogenous.

Further research may be necessary to explore how to address the endogeneity of policy formulation.

Thanks, everyone!